Limit distribution theory for maximum likelihood estimation of a log-concave density
Fadoua Balabdaoui, Kaspar Rufibach, Jon A. Wellner
The Annals of Statistics · 2009 · Volume 37 · Issue 3
Abstract
We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0=exp ϕ0 where ϕ0 is a concave function on ℝ. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0=log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.
Keywords
62G05, 62G20, 62N01, asymptotic distribution, integral of Brownian motion, invelope process, log-concave density estimation, lower bounds, maximum likelihood, mode estimation, nonparametric estimation, qualitative assumptions, shape constraints, strongly unimodal, unimodal
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